Theorem bj-alequex 32708

Description: A fol lemma. See bj-alequexv 32655 for a version with a DV condition requiring fewer axioms. Can be used to reduce the proof of spimt 2253 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-alequex	|- ( A. x ( x = y -> ph ) -> E. x ph )

Proof of Theorem bj-alequex

Step	Hyp	Ref	Expression
1		ax6e 2250	. 2 |- E. x x = y
2		exim 1761	. 2 |- ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ph ) )
3	1, 2	mpi 20	1 |- ( A. x ( x = y -> ph ) -> E. x ph )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bj-spimt2  32709  bj-equsal1t  32809